Had a bathroom remodel about 2 years ago. Its one of those toilets with the option to flush using less water and then another for a full one. I've noticed recently when flushing the full one, the toilet gasps for air as and after the water goes down. It's loud, sounds like it is choking.

I think this means a clogged vent pipe? Also red where it is a clog in the line, but this has been going on a long time now and I would have thought the clog would have stopped all flow by now. What is the best way to DIY this?

Get on the roof and look down the vent pipe. Maybe bring a garden hose and see if it fills with water.

if I run something down the stack don't I risk pushing the problem further down the line? Would prefer to lift it out

My toilet made the same noise when I had a piece of plastic stuck in it. Had to remove the toilet last week to fish it out.